Hypermassive Ordinal
An ordinal \(\alpha\) is a \(\hat{\psi}\) Function fixed point if it is a fixed point of \( \alpha \mapsto \hat{\psi} (\alpha) \). The \(\beta\)-th \(\hat{\psi}\) Function fixed point is denoted as \(\hat{\psi}_{\beta}\). If \(\beta\) is uncountable or greater, then \(\hat{\psi}_{\beta}=\hat{\psi}_{\hat{\psi}(\beta)}\). Define a variant of the \(\hat{\psi}\) function: The \(_{2}\hat{\psi}\) function. The \(_{2}\hat{\psi}\) function is similar to the \(\hat{\psi}\) function, except the set \(C_{0}\) in the definition contains all \(\hat{\psi}\) function fixed points as well, plus the recursive rule contains a function \(\alpha \mapsto \hat{\psi}_{\alpha+1}\) (?). Better definition? \( C_{0}(\gamma,\delta) = \{0,1,\omega,\Omega\} \cup \bigcup_{\varepsilon < \gamma}{\hat{\phi}(\varepsilon ; 1)} \\ C_{n+1}(\gamma,\delta) = \{ \zeta + \eta, \zeta \eta, \zeta^{\eta}, \hat{\psi}(\iota) : \zeta, \eta, \iota \in C_{n}(\gamma,\delta) : \iota < \delta \} \\ C(\gamma,\delta)=\bigcup_{n<\omega}{C_{n}(\gamma,\delta)} \\ \hat{\psi}(\alpha,\beta) = \min\{\iota < \hat{\phi}(\alpha ; 1) : \iota \notin C(\alpha,\beta)\} \\ \hat{\psi}_{C}1(\beta) = \hat{\psi}(1,\beta) \\ \hat{\psi}_{C}\alpha(\beta) = \hat{\psi}(\alpha,{\hat{\phi}(\alpha ;1)}\uparrow {\hat{\psi}(\alpha-1,\beta)}) \\ \hat{\psi}_{C}\alpha(\beta) = \sup\{\hat{\psi}(\alpha,\hat{\psi}(\gamma,\beta)):\gamma <\alpha\} \text{ if }\alpha \text{ is limit} \\ \hat{\psi}_{C}(\beta) = \hat{\psi}_{C} \delta:\hat{\phi}(\delta ;1)<\beta \} (\beta) \\ \hat{\psi}_{C}(\beta) = \hat{\psi}_{C} \delta:\hat{\phi}(\delta ;1)>\varepsilon \} : \varepsilon < \beta \} (\beta) \text{ if } min\{ \delta:\hat{\phi}(\delta ;1)>\beta \} \text{ is limit} \\ \hat{\psi}(\alpha) = \hat{\psi}_{C}(\alpha) \) ^^^ Normal \(\hat{\psi}\) ^^^ \( A_{0} = \bigcup_{n < \omega}{\hat{\psi}_{n}} \\ A_{n+1} = \bigcup_{\vartheta \in A_{n} }{\hat{\psi}_{\vartheta}} \\ A = \bigcup_{n < \omega}{A_{n}} \\ C_{0}(\gamma,\delta) = \{0,1,\omega,\Omega\} \cup \bigcup_{\varepsilon < \gamma}{\hat{\phi}(\varepsilon ; 1)} \cup A \\ C_{n+1}(\gamma,\delta) = \{ \zeta + \eta, \zeta \eta, \zeta^{\eta}, \hat{\psi}(\iota), {_{2}}\hat{\psi}(\iota), \hat{\psi}_{\iota+1} : \zeta, \eta, \iota \in C_{n}(\gamma,\delta) : \iota < \delta \} \\ C(\gamma,\delta)=\bigcup_{n<\omega}{C_{n}(\gamma,\delta)} \\ {_{2}}\hat{\psi}(\alpha,\beta) = \min\{\iota < \hat{\phi}(\alpha ; 1) : \iota \notin C(\alpha,\beta)\} \\ {_{2}}\hat{\psi}_{C}1(\beta) = {_{2}}\hat{\psi}(1,\beta) \\ {_{2}}\hat{\psi}_{C}\alpha(\beta) = {_{2}}\hat{\psi}(\alpha,{\hat{\phi}(\alpha ;1)}\uparrow \hat{\psi}(\alpha-1,\beta)}) \\ {_{2}}\hat{\psi}_{C}\alpha(\beta) = \sup\{_{2}\hat{\psi}(\alpha,{_{2}}\hat{\psi}(\gamma,\beta)):\gamma <\alpha\} \text{ if }\alpha \text{ is limit} \\ _{2}\hat{\psi}_{C}(\beta) = {_{2}}\hat{\psi}_{C} \delta:\hat{\phi}(\delta ;1)<\beta \} (\beta) \\ {_{2}}\hat{\psi}(\alpha) = {_{2}}\hat{\psi}_{C}(\alpha) \) ^^^ _2 \(\hat{\psi}\) ^^^ Fixed points of \({_{2}}\hat{\psi}\) are demoted using \({_{2}}\hat{\psi}_{\alpha}\). \({_{3}}\hat{\psi}\) can be created similarly, as with \({_{\omega}}\hat{\psi}\), etc. Attempted definition of _n \(\hat{\psi}\) for n>2: \( A_{\mu}(0) = \bigcup_{n < \omega}\{\hat{\psi}_{n,\mu-1}\} \\ A_{\mu}(0) = \bigcup_{n < \omega} \{\sup \{\hat{\psi}_{n,\nu} : \nu < \mu \} \} \text{ if } \mu \text{ is limit} \\ A_{\mu}(n+1) = \bigcup_{\vartheta \in A_{\mu}(n) }{\hat{\psi}_{\vartheta,\mu}} \\ A = \bigcup_{n < \omega} \{A_{\mu}(n)\} \\ C_{0}(\gamma,\delta,\mu) = \{0,1,\omega,\Omega\} \cup \bigcup_{\varepsilon < \gamma} \{\hat{\phi}(\varepsilon ; 1)\} \cup A_{\mu} \\ C_{n+1}(\gamma,\delta,\mu) = \{ \zeta + \eta, \zeta \eta, \zeta^{\eta}, \hat{\psi}(\iota), {_{\nu}}\hat{\psi}(\iota), {_{\rho}}\hat{\psi}_{\iota+1} : \zeta, \eta, \iota \in C_{n}(\gamma,\delta,\mu) ; \iota < \delta ; \rho < \nu \le \mu \} \\ C(\gamma,\delta,\mu)=\bigcup_{n < \omega} \{C_{n}(\gamma,\delta,\mu)\} \\ {_{\mu}}\hat{\psi}(\alpha,\beta) = \min\{\iota < \hat{\phi}(\alpha ; 1) : \iota \notin C(\alpha,\beta,\mu)\} \\ {_{\mu}}\hat{\psi}_{C}1(\beta) = {_{\mu}}\hat{\psi}(1,\beta) \\ {_{\mu}}\hat{\psi}_{C}\alpha(\beta) = {_{\mu}}\hat{\psi}(\alpha,{\hat{\phi}(\alpha ;1)}\uparrow \hat{\psi}(\alpha-1,\beta)}) \\ {_{\mu}}\hat{\psi}_{C}\alpha(\beta) = \sup\{_{\mu}\hat{\psi}(\alpha,{_{\mu}}\hat{\psi}(\gamma,\beta)):\gamma <\alpha\} \text{ if } \alpha \text{ is limit} \\ _{\mu}\hat{\psi}_{C}(\beta) = {_{\mu}}\hat{\psi}_{C} \delta:\hat{\phi}(\delta ;1)<\beta \} (\beta) \\ {_{\mu}}\hat{\psi}(\alpha) = {_{\mu}}\hat{\psi}_{C}(\alpha) \\ \hat{\psi}_{\alpha,\beta}={_{\beta}}{\hat{\psi}}_{\alpha} \\ {\hat{\psi}}_{0,\beta}=0 \\ {\hat{\psi}}_{\alpha,\beta} = {_{\beta}}\hat{\psi} (\hat{\psi}_{\alpha,\beta}) \\ {\hat{\psi}}_{\alpha,\beta} \ge {\hat{\psi}}_{\gamma,\beta} \Leftrightarrow \alpha \ge \gamma \\ {\hat{\psi}}_{\alpha,\beta} = \sup \{ {\hat{\psi}}_{\gamma,\beta} : \gamma < \alpha \} \text{ if } \alpha \text{ is limit} \) Note that \({_{1}}\hat{\psi}(\alpha)=\hat{\psi}(\alpha)\). More Fixed Points \( _{\beta}{\hat{\psi}}_{\alpha} \) is the \(\alpha\)-th fixed point of the \(\beta\)-th \(\hat{\psi}\) function, which can also be denoted as \( {\hat{\psi}}_{\alpha,\beta} \). If \(\alpha\) is limit, \( {\hat{\psi}}_{\alpha,\beta} = \sup\ _{\alpha,\gamma} : \gamma<\beta\} \) If \(\beta\) is limit, \( {\hat{\psi}}_{\alpha,\beta} = \sup\ _{\gamma,\beta} : \gamma<\alpha\} \) Unofficial, ill-defined, no definitions yet: Adding more comma-separated arguments to the subscript will recurse over each old function. For example: The \(\hat{\psi}_{\cdots,2}\) function is similar to the \(\hat{\psi}_{\cdots}\) function, except the set \(C_{0}\) in the definition contains all \(\hat{\psi}_{\cdots}\) function fixed points as well, plus the recursive rule contains a function \(\alpha \mapsto \hat{\psi}_{\alpha+1,\cdots}\), where \(\cdots\) represents the rest of the array. WIP Category:ORDINALS Category:TRANSFINITE NUMBERS Category:INFINITY Category:SET THEORY Category:C7X's Stuff Category:CLEVER STUFF